Question: Determine how many solutions exist for the system of equations. ${-15x+3y = 30}$ ${9x+3y = 15}$
Answer: Convert both equations to slope-intercept form: ${-15x+3y = 30}$ $-15x{+15x} + 3y = 30{+15x}$ $3y = 30+15x$ $y = 10+5x$ ${y = 5x+10}$ ${9x+3y = 15}$ $9x{-9x} + 3y = 15{-9x}$ $3y = 15-9x$ $y = 5-3x$ ${y = -3x+5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x+10}$ ${y = -3x+5}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.